This paper presents a trajectory generation algorithm that efficiently computes high-performance flight trajectories that are capable of moving a quadrocopter from a large class of initial states to a given target point that will be reached at rest. The approach consists of planning separate trajectories in each of the three translational degrees of freedom, and ensuring feasibility by deriving decoupled constraints for each degree of freedom through approximations that preserve feasibility. The presented algorithm can compute a feasible trajectory within tens of microseconds on a laptop computer; remaining computation time can be used to iteratively improve the trajectory. By replanning the trajectory at a high rate, the trajectory generator can be used as an implicit feedback law similar to model predictive control. The solutions generated by the algorithm are analyzed by comparing them with time-optimal motions, and experimental results validate the approach.

A method is presented for the rapid generation and feasibility verification of motion primitives for quadrocopters and similar multirotor vehicles. The motion primitives are defined by the quadrocopter's initial state, the desired motion duration, and any combination of components of the quadrocopter's position, velocity, and acceleration at the motion's end. Closed-form solutions for the primitives are given, which minimize a cost function related to input aggressiveness. Computationally efficient tests are presented to allow for rapid feasibility verification. Conditions are given under which the existence of feasible primitives can be guaranteed a priori . The algorithm may be incorporated in a high-level trajectory generator, which can then rapidly search over a large number of motion primitives which would achieve some given high-level goal. It is shown that a million motion primitives may be evaluated and compared per second on a standard laptop computer. The motion primitive generation algorithm is experimentally demonstrated by tasking a quadrocopter with an attached net to catch a thrown ball, evaluating thousands of different possible motions to catch the ball.

We present the design of a system that allows quadrocopters to balance an inverted pendulum, throw it into the air, and catch and balance it again on a second vehicle. Based on first principles models, a launch condition for the pole is derived and used to design an optimal trajectory to throw the pole towards a second quadrocopter. An optimal catching instant is derived and the corresponding position is predicted by simulating the current position and velocity estimates forward in time. An algorithm is introduced that generates a trajectory for moving the catching vehicle to the predicted catching point in real time. By evaluating the pole state after the impact, an adaptation strategy adapts the catch maneuver such that the pole rotates into the upright equilibrium by itself. Experimental results demonstrate the performance of the system.

Quadrocopters exhibit complex high-speed fight dynamics, and the accurate modeling of these dynamics has proven diffcult. Due to the use of simplifed models in the design of feedback control algorithms, the execution of highperformance fight maneuvers under pure feedback control typically leads to large tracking errors. This paper investigates an iterative learning scheme aimed at the non-causal compensation of repeatable trajectory tracking errors over the course of multiple executions of periodic maneuvers. The learning is carried out in the frequency domain and uses a simplifed model of the closed-loop dynamics of quadrocopter and feedback controller. The resulting algorithm requires little computational power and memory, and its convergence is shown for the nominal model. This paper further introduces a time-scaling method that allows the initial learning to occur at reduced speeds, thus extending the applicability of the algorithm for high performance maneuvers. The presented algorithms are validated in experiments, with a quadrocopter fying a fgure-eight maneuver at high speed.

An algorithm is proposed allowing for the rapid generation and evaluation of quadrocopter state interception trajectories. These trajectories are from arbitrary initial states to final states defined by the vehicle position, velocity and acceleration with a specified end of time. Sufficient criteria are then derived allowing trajectories to be tested for feasibility with respect to thrust and body rates. It is also shown that the range of a linear combination of the vehicle state can be solved for in closed form, useful e.g. for testing that the position remains within a box. The algorithm is applied by revisiting the problem of finding a trajectory to hit a ball towards a target with a racket attached to a quadrocopter. The trajectory generator is used in a model predictive control like strategy, where thousands of trajectories are generated and evaluated at every controller update step, with the first input of the optimal trajectory being sent to the vehicle. It is shown that the method can generate and evaluate on the order of one million trajectories per second on a standard laptop computer.

This paper presents an algorithm that permits the calculation of interception maneuvers for quadrocopters. The translational degrees of freedom of the quadrocopter are decoupled. Pontryagin's minimum principle is used to show that the interception maneuver that minimizes the time to rest after the interception is identical to the time-optimal maneuver that drives the vehicle to the position at which it comes to rest after the interception. This fact is leveraged to apply previously developed, computationally efficient methods for the computation of interception maneuvers. The resulting trajectory generation algorithm is computationally lightweight, permitting its use as an implicit feedback law by replanning the trajectory at each controller update. The validity and performance of the approach is demonstrated experimentally by intercepting balls mid-flight. The real-time trajectory generation permits to take into account changes in the predicted ball flight path at each controller update.

This paper presents a method for enabling a fleet of circularly arranged quadrocopters to throw and catch balls with a net. Based on a first-principles model of the net forces, nominal inputs for all involved vehicles are derived for arbitrary target trajectories of the net. Two algorithms that generate open-loop trajectories for throwing and catching a ball are also introduced. A set of throws and catches is demonstrated in the ETH Zurich Flying Machine Arena testbed.

We extend the classic control problem of the inverted pendulum by placing the pendulum on top of a quadrotor aerial vehicle. Both static and dynamic equilibria of the system are investigated to find nominal states of the system at standstill and on circular trajectories. Control laws are designed around these nominal trajectories. A yaw-independent description of quadrotor dynamics is introduced, using a `Virtual Body Frame'. This allows for the time-invariant description of curved trajectories. The balancing performance of the controller is demonstrated in the ETH Zurich Flying Machine Arena testbed. Development potential for the future is highlighted, with a focus on applying learning methodology to increase performance by eliminating systematic errors that were seen in experiments.

The Flying Machine Arena (FMA) is an indoor research space built specifically for the study of autonomous systems and aerial robotics. In this video, we give an overview of this testbed and some of its capabilities. We show the FMA infrastructure and hardware, which includes a fleet of quadrocopters and a motion capture system for vehicle localization. The physical components of the FMA are complemented by specialized software tools and components that facilitate the use of the space and provide a unified framework for communication and control. The flexibility and modularity of the experimental platform is highlighted by various research projects and demonstrations.

A numerical method for computing quadrocopter maneuvers between two states is presented. Computed maneuvers satisfy Pontryagin's minimum principle with respect to time-optimality. First, in order to obtain the structure of time-optimal maneuvers, we apply the minimum principle to a first-principles, two-dimensional quadrotor model. Then we present a numerical algorithm that enables the computation of maneuvers for arbitrary initial and final states. The developed method is used to compute a set of maneuvers, which are discussed and demonstrated experimentally in the ETH Zurich Flying Machine Arena testbed.